Question

A shopkeeper sold an article at a gain of 5%. If he had sold it for Rs 16.50 less, he would have lost 5%. Find the cost price of the article.

Answer 1

Hint: Assume the cost price of the article is x. Find the selling price using \[gain\%=\dfrac{S.P-C.P}{C.P}\times 100\]
Use the relation $loss\%=\dfrac{C.P-S.P}{C.P}\times 100$. Use these two relations to get an equation in x. Solve for x to get the cost price of the article.

Complete step-by-step answer:

Let x be the cost price of the article.

Using \[gain%=\dfrac{S.P-C.P}{C.P}\times 100\] and substituting gain% = 5, we get
$5=\dfrac{S.P-x}{x}\times 100$

Multiplying both sides by x, we get
5x=100S.P-100x

Adding 100x on both sides, we get
5x+100x=100 S.P-100x+100x
i.e. 105x = 100S.P

Dividing both sides by 100 we get
1.05x = S.P
i.e. S.P = 1.05x

When a loss of 5% occurs selling price = 1.05x-16.5

Using $loss%=\dfrac{C.P-S.P}{C.P}\times 100$ and substituting loss = 5, S.P = 1.05x-16.5, we get
$\begin{align}
  & 5=\dfrac{x-\left( 1.05x-16.5 \right)}{x}\times 100 \\
 & \Rightarrow 5=\dfrac{x-1.05x+16.5}{x}\times 100 \\
 & \Rightarrow 5=\dfrac{16.5-0.05x}{x}\times 100 \\
\end{align}$

Multiplying both sides by x we get
5x= 1650-5x
Adding 5x on both sides, we get
5x+5x = 1650
i.e. 10x=1650

Dividing both sides by 10, we get
$\dfrac{10x}{10}=\dfrac{1650}{10}$
i.e. x = 165

Hence the cost price of the item = Rs 165

Note: [1] The formula for gain% is $\dfrac{gain}{C.P}\times 100$ , but since gain = S.P -C.P we have \[gain\%=\dfrac{S.P-C.P}{C.P}\times 100\]
Similarly, the formula for $loss\%$ is $\dfrac{loss}{C.P}\times 100$, but since loss = C.P -S.P we have
$loss\%=\dfrac{C.P-S.P}{C.P}\times 100$
[2] We could have also solved the question by finding S.P in case of gain and S.P in case of loss in terms of x and then used the relation, ${S.P}_{gain}={ S.P}_{loss} +16.50$